Another Proof for the Continuity of the Lipsman Mapping
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ANOTHER PROOF FOR THE CONTINUITY OF THE LIPSMAN MAPPING A. Messaoud and A. Rahali
UDC 515.1
We consider a semidirect product G = K n V where K is a connected compact Lie group acting b by automorphisms on a finite-dimensional real vector space V equipped with an inner product h, i. By G we denote the unitary dual of G and by g‡ /G we denote the space of admissible coadjoint orbits, where b and g‡ /G g is the Lie algebra of G. It was indicated by Lipsman that the correspondence between G is bijective. Under certain assumptions on G, we give another proof of continuity for the orbit mapping (Lipsman mapping) b ⇥ : g‡ /G −! G.
1. Introduction b be the unitary dual of G, i.e., the set of all Let G be a second countable locally compact group and let G b is equipped with the Fell equivalence classes of irreducible unitary representations of G. It is well known that G topology [6]. The description of the dual topology is a good candidate for some aspects of harmonic analysis on G (see, e.g., [4, 5]). For a simply connected nilpotent Lie group and, more generally, for an exponential solvable Lie b is homeomorphic to the space of coadjoint orbits g⇤ /G through the Kirillov group G = exp(g), its dual space G mapping (see [8]). In the context of semidirect products G = K n N of compact connected Lie groups K acting upon simply connected nilpotent Lie groups N, it was pointed out by Lipsman [9] that we again have an orbit picture of the space dual to G. The unitary dual space of Euclidean groups of motion is homeomorphic to the admissible coadjoint orbits [5]. This result was generalized in [4] for a class of Cartan groups of motion. In the present paper, we consider the semidirect product G = K n V, where K is a connected compact Lie group acting by automorphisms upon a finite-dimensional real vector space V equipped with an inner product h, i. In the spirit of the orbit method due to Kirillov, Lipsman established a bijection between a class of coadjoint orbits b For every admissible linear form of G and the unitary dual G. of the Lie algebra g of G, we can construct an irreducible unitary representation ⇡ by holomorphic induction and, according to Lipsman (see [9]), every irreducible representation of G arises in this way. Then we get a map from the set g‡ of admissible linear forms b of G. Note that ⇡ is equivalent to ⇡ 0 if and only if and 0 are on the same G-orbit. onto the dual space G b Finally we obtain a bijection between the space g‡ /G of admissible coadjoint orbits and the unitary dual G. Definition 1. Let G be a (real) Lie group, let g be its Lie algebra, and let exp : g −! G be its exponential map. We say that G is exponential if exp(g) = G. We now formulate the main result of the present paper, i.e., another proof for the continuity of the orbit mapping (see [11]): University of Sfax, Sfax, Tunisia; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 945–951, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.548. Original article submitted May 10, 2017. 1100
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